10 SPACES WITH OPERATORS. ORBIT SPACES 45 (metrizable) topology induced on G by that of ^R(I) is compatible with the group structure of G, and that right Cauchy sequences in G are the same as Cauchy sequences (in G) with respect to the norm on ^R(I) (7.1). Give an example of a right Cauchy sequence in G which is not a left Cauchy sequence. 9. (a) Let G' be a complete metrizable group and let G0 ^ G' be a dense subgroup of G'. Let G be the topplogical group obtained by giving G0 the discrete topology. The identity mapping G -> G0 is a continuous bijective homomorphism, but its extension by continuity to a homomorphism G -> G' (1 2.9.4) is not surjective. (b) Let G be the dense subgroup Q2 of R2, let 6 be an irrational number, and let u be the continuous homomorphism (x, y)\- >x + By of G into R. Let G' = w(G). The mapping #, considered as a homomorphism of G into G', is bijective, but its continuous extension to a homomorphism of R2 into R is not injective. (c) Use (a) and (b) to construct an example of two complete groups Gj, G2 and a continuous bijective homomorphism 'u of a dense subgroup H! of Gx onto a dense subgroup H2 of G2 , such that the continuous extension of u to a homomorphism of Gi into G2 is neither injective nor surjective. 10. Let / be a continuous homomorphism of a subgroup H ^ {0} of R into a locally compact metrizable group G. Show that if /is not an isomorphism of H onto the subgroup /(H) of G, then /(H) is relatively compact in G. (Reduce to the case where H is closed in R and/(H) is dense in G. Begin by showing that, for each neighborhood W of the neutral element e in G, the set/-1(W) is unbounded, and deduce that /(H n R+) is dense in G. If V is a compact symmetric neighborhood of e in G, show that there exist a finite number of elements tt > 0 in H such that the neighborhoods /(/,)V cover V. For each x e G, let A* be the set of all t e H such that /(/) e xV. Show that, if t e A* , there exists a tt such that / — /, e Ax , and deduce that, if I is the largest compact interval with origin 0 in R containing all the // , then I n Ax is not empty. Deduce that G C/(I) • V is compact.) 11. Let G be a commutative metrizable topological group, and let d be an invariant distance defining the topology of G. Let h be the HausdorrT distance on 3f(G) cor- responding to d (Section 3.16, Problem 3). Show that if M, N, P, Q are four bounded closed subsets of G, then /z(MP, NQ) <J h(M, N) + A(P, Q). 10. SPACES WITH OPERATORS. ORBIT SPACES Let G be a group and E a set. An action (or left action) of G on E is a mapping (s, x)t->s - x of G x E into E satisfying the following conditions: (1) If e is the neutral element of G, then e - x = x for all x e E. (2) For all s, t in G we have s - (t - x) = (st) - x for all x e E. These conditions imply that s"1 • (s • x) = x for all s e G and all xeE; hence for each s e G the mapping x\-*s • x is a bijection of E onto E, and the inverse bijection is x\-*s~l • x. For each x e E, the set G • x of elements s • x where s e G is called the orbit of x (for the given action of G on E). The set S* of elements s e G suchem